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Физико-математические науки
NONLINEAR WAVES IN A CYLINDER SHELL CONTAINING A VISCOUS LIQUID WITH INERTIA OF ITS MOVEMENT, UNDER THE IMPACT OF THE SURROUNDING ELASTIC MEDIUM AND STRUCTURAL DAMPING IN THE LONGITUDINAL DIRECTION
Evdokimova E.V. 2, Kondratov D.V. 3, Blinkov Y.A. 4, Mogilevich L.I. 1

1. Yuri Gagarin State Technical University of Saratov
2. Yuri Gagarin state technical university of Saratov
3. Volga Management Institute named after P.А. Stolypin - a branch of Federal State-Funded Educational Institution of Higher Education Russian Presidential Academy of National Economy and Public Administration
4. Saratov State University

Abstract:

The present article deals with further developing of perturbation method for deformation non-linear waves in an elastic cylinder shell, filled with viscous incompressible liquid without inertia of its movement, surrounded by an elastic media and under construction damping in longitudial direction. Surrounding medium presence leads to integral-differential equation, to generalizing Korteweg-de Vries ones and possessing the same soliton in the form of a solitary wave - a soliton. It does not contain an arbitrary constant number unlike Korteweg-de Vries equation solution. The viscous incompressible liquid presence inside the shell behavior is described by means of dynamics and continuity equation, is solved together with boundary conditions liquid adhesion to a shell wall. The solution is presented by direct expansion of unknown function by small parameter of hydroelasticity problem and reduced to the problem for hydrodynamics lubrication theory equations. The equations solution defines the tensions on the part of the liquid, the tensions influence the shell longitudinal and normal directions. The liquid presence in the shell adds to longitudial deformation waves equations one more equation member, which does not allow to find exact solution. Construction damping in a longitudial direction adds the same equation member, like liquid presence does. They posses opposite signs in the case of shell Poisson coefficient being smaller than 1/2. In contrary case signs coincide. Liquid presence in the shell and construction damping demand for numerical research. The liquid presence leads to the equation, generalizing Kortevega-de-Vrisa equation, lacking the exact solution and demanding numerical investigation. The numerical investigation is carried out with the use of the modern approach, relying on the universal algorithm of commutative algebra for integro-interpolation method. As a result of difference Gröbner basis construction, the difference Crank-Nicolson type schemes are generalized. The schemes were obtained due to the use of basic integral difference correlations, approximating the initial equations system. Computational experiment showed that in the case of construction damping and liquid impact have opposite signs but coincide in value, their influence does not case and a soliton propagates without changing its direction and its amplitude, which coincide analytical solution. If constructive damping exceeds liquid impact, the wave amplitude decreases; in the opposite case the wave amplitude grouse.

Keywords: elastic cylinder shell, viscous incompressible liquid, nonlinear waves

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